Unconventional acoustic wave propagation transitions induced by resonant scatterers in the high-density limit

Experiments on ultrasound propagation through a gel doped with resonant encapsulated microbubbles provided evidence for a discontinuous transition between wave propagation regimes at a critical excitation frequency. Such behavior is unlike that observed for soft materials doped with non-resonant air or through liquid foams, and disagrees with a simple mixture model for the effective sound speed. Here, we study the discontinuous transition by measuring the transition as a function of encapsulated microbubble volume fraction. The results show the transition always occurs in the strong-scattering limit (l/λ < 1, l and λ are the mean free path and wavelength, respectively), that at the critical frequency the effective phase velocity changes discontinuously to a constant value with increasing microbubble volume fraction, and the measured critical frequency shows a power law dependence on microbubble volume fraction. The results cannot be explained by multiple scattering theory, viscous effects, mode decoupling, or a critical density of states. It is hypothesized the transition depends upon the microbubble on-resonance effective properties, and we discuss the results within the context of percolation theory. The results shed light on the discontinuous transition’s physics, and suggest soft materials can be engineered in this manner to achieve a broad range of physical properties with potential application in ultrasonic actuators and switches.


Supplementary Note 1: EMB resonance frequency, scattering cross section, and damping
The encapsulated microbubble (EMB) linear harmonic oscillation is obtained from solutions of a Rayleigh-Plesset-like differential equation [S1], which considers damping due to acoustic radiation and the EMB shell viscosity.The EMB resonance frequency fO and angular frequency wO = 2pfO prediction takes into account both the EMB shell and suspending gel densities (rS = 1,184 kg/m 3 for the EMB shell, and the gel density was determined separately for each sample prior to adding the dopants so that the EMB volume fraction f could be obtained; for example, for f = 0.81% the starting gel density was determined to be rl = 995.0kg/m 3 ± 0.2 kg/m 3 ), the shell inner and outer radii (a shell thickness e = 100 nm is assumed based upon prior scanning electron microscopy measurements [S2]), the gel hydrostatic pressure (taken to be P =101.325kPa), the enclosed gas polytropic index (g ~ 1.1), and the shell shear modulus (GS ~1.43 GPa) that was studied in prior work [S3].The resonance angular frequency can be expressed in terms of the EMB equilibrium diameter DO as than the e 3 term, while in the denominator the e term is over three orders of magnitude larger than the e 2 term and over six orders of magnitude larger than the e 3 term.
In the limit that e goes to zero, we recover the Minnaert formula for a gas bubble [S4]: for DO = 90 µm, the Minnaert formula predicts fO = 64,850 Hz, which is far below the experiment's frequency range and does not explain the sound transmission properties of our samples between 50-800 kHz.However, to first order in e the resonance frequency can be written in terms of the EMB equilibrium radius RO as This expression yields fO = 695,055 Hz for DO = 90 µm, which is in agreement with the experimental results.Thus, to leading order in e, the resonance frequency is shifted by the two new terms 12GSe and e(rS-3rl) resulting in a modified Minnaert formula due to the suspending gel and EMB shell properties.
For a specified EMB diameter the w-dependent scattering cross section s(w) is given by where , R10 and R20 are the inner and outer EMB equilibrium radii, respectively, and b is the total damping coefficient.Damping from acoustic radiation bAR is determined by where vO is take to be the measured (effective) phase velocity for each sample (i.e., for each value of f).
Supplementary Fig. 2 shows the predicted s(f) for a single, isolated EMB with DO = 90 µm (normalized to the equilibrium scattering cross section), which takes into account acoustic radiation damping.On resonance, the effective scattering cross section is several orders of magnitude larger than the equilibrium scattering cross section.Further, the inset to Supplementary Fig. 2 shows that acoustic radiation damping dominates the damping coefficient within the frequency range of interest, which is where the EMB resonance frequency range overlaps the experimental frequency range (primarily, 400-800 kHz).Damping from the EMB shell viscosity bSV is determined by where µS ~ 10 Pa*s is the estimated PAN shell shear viscosity [S5].The thermal damping coefficient for EMB oscillations [S6] is determined from where DP = (rl -rS)/rS, Z is the gas Peclet number (for isopentane we use rg = 616 kg/m 3 , cg = 1,660 J/kg*K, and kg = 0.11 W/m*K), and G± is given by Lastly, in computations of s(w), bAR, and the angular frequency dependent mean free path l(w) it is useful to know the value of DO corresponding to each EMB wO.We solve for DO as a function of wO by solving the following cubic equation, which follows from the modified Minnaert formula keeping only terms to leading order in e:

Supplementary Note 2: The density of states and the density of active oscillators
The number of EMBs on resonance at the resonance frequency fO can be expressed as where  345 !"! is the total number of EMBs within a sample (i.e., for each value of EMB equilibrium volume fraction f).Through knowledge of the relationship between DO and fO (Supplementary Note 1) we can also write this expression as where "Gaussian distribution(DO)" is the solid red line shown in the main text's Fig. 1c (thus, in computing the density of states nw we use the normalized Gaussian fit to the main text's Fig. 1c data).To compute  345 ( ! ) and therefore  345 ( ! ) it is necessary to determine  345 !"! for each f value.If all EMBs within a sample had the same diameter then the total EMB volume  345 !"! is simply where V1EMB is the volume of a single EMB and V is the overall sample volume.In this case, since V1EMB, f, and V are known quantities this expression would give a simple way of determining  345 !"! :  345 !"! =   &345 ⁄ .However, not all EMBs in our samples have the same diameter, and so the measured distribution shown in the main text's Fig. 1c must be taken into account in determining  345 !"! .In a similar manner, we compute  345 !"! through where the upper and lower integral limits are taken to be D1 = 1 µm and D2 = 140 µm (corresponding to the lower and upper ranges of the measured EMB size distribution), and With  345 !"! known we can then compute  345 ( ! ) and therefore  345 ( ! ) using the above expressions and relationship between DO and fO, which gives the number of EMBs at each fO for a given value of f.
To compute the density of states nw (i.e., the EMB density per Hz) we divide  345 ( ! ) by the factor 2pV(dfO) (i.e.,  7 =  345 ( ! ) &2( !)( ⁄ ) where the quantity dfO is computed by determining dfO/dRO from Eq. S3 in Supplementary Note 1 which yields Upon computing dfO/dRO we multiple by dRO in order to find dfO, where dRO = 10 µm is from the measured EMB count versus DO distribution bin size where the uncertainty in the EMB radius is ± 5 microns, which gives a total dRO = 10 µm.Supplementary Figure 3 shows the density of states nw versus resonance angular frequency wO squared for multiple EMB equilibrium volume fractions f for which data is presented within the main text.In our experiments, nw versus wO 2 is a one-sided, non-Gaussian distribution, which peaks at f = 674 kHz.That the stiffness K = mwO 2 where m is the oscillator mass (which varies weakly across the distribution) implies nw versus K is also non-Gaussian.
The density of active oscillators ractive (EMBs m -3 ) contributing to the scattering at each frequency f can be found from the expression where bAR is the acoustic radiation damping factor given by Eq.S5 in Supplementary Note 1 and the quantity 2bAR corresponds to the full-width at half-maximum of the scattering cross section (as exemplified by Supplementary Fig. 2). Figure 2c  where b is the radiative damping factor and  %$ = 4 "' ⁄ ], rl and rS are the suspending gel and EMB shell densities, respectively, and R10 and R20 are the inner and outer EMB equilibrium radii, respectively.
Rearranging the integral's denominator and factoring out an w 2 from each term in square brackets yields Making the substitution y = wO 2 /w 2 , and for the resonance condition w = wO, the integral becomes By assuming negligible contribution to the integral for −∞ <  < 0 we can extend the limits of integration to ±∞ and integrate over the complex plane, which yields the expression where the quantity nw has been found by evaluating nOw at w.
In computing l -1 (w), b is dominated by acoustic radiation damping with the damping coefficient bAR provided in Supplementary Note 1 (Eq.S5).Values for swO are computed knowing R10 and R20 as a function of frequency, which is determined by solving the cubic equation (Eq.S9) discussed in Supplementary Note 1 that considers all terms to first order in for a finite seff(f) peak width in our computations of Deff we determine the average effective scattering cross section seff_AVG(f) and then from seff_AVG(f) we compute an average effective diameter Deff_AVG(f).Integrating over the total scattering cross section in the effective medium (using the measured wave speed, as opposed to the undoped gel wave speed) gives which yields seff(f)/seff_AVG(f) = 4/p, and subsequently

Values for
Deff_AVG(f) are used to determine the effective volume fraction feff.
The predictions shown in Supplementary Fig. 9 highlight that for a single EMB on resonance Deff is significantly larger than DO (between a factor of 15 and 19 greater depending upon the frequency).The predictions shown in Supplementary Fig. 10 highlight that in this case Lastly, Supplementary Fig. 11 shows that within the range of f over which the ISA is applicable, feff varies considerably (from 11.5% up to 27.5%), which explains how such effects are observable in our samples despite the low (a few percent) EMB equilibrium volume fractions.Also, the peak in feff versus f near f = 550 kHz is the result of an increasing EMB density of active oscillators ractive(f) and simultaneous decrease in Deff(f) with increasing frequency.
We do not expect absorption to skew our late-time analysis because the characteristic absorption time ta ≥ 100 µs is an order of magnitude larger than tD (which is found by multiplying T(t) within the SCT by the factor  +(; H & ⁄ ) ).Further, taking ta ~tD = 10 µs cannot account for the observed late-time deviations from linearity, and this is demonstrated by the dashed purple lines in Supplementary Fig. 15 for representative f values.We conclude that absorption is unimportant in our measurements.Further, fO = 800 kHz (the maximum frequency in our experiments) corresponds to a resonating EMB with DO = 82 µm, which is near the maximum in the EMB size distribution shown in the main text's Fig. 1c.Yet, despite the continuous increase in the density of active oscillators with increasing frequency the loss of the incoherent signal for f > fC* further supports the conclusion that absorption does not skew our late-time analysis since we would expect an increasing scatterer density might lead to higher absorption levels and persistent linear late-time behavior (which is not observed experimentally).
within the main text shows a representative plot of ractive versus f for f = 0.21%, which highlights how the number of EMBs contributing to the scattering in our samples (i.e., the disorder strength) increases with increasing frequency.SupplementaryNote 3: Measured change in phase angle, longitudinal phase velocity, effective wavelength, and scattering mean free pathThe longitudinal phase velocity vL, determined from the coherent part of the transmitted wave, is found from the change in phase angle Dq as a function of f (see Fig.3of the main text for examples) where the phase spectrum is extracted from the transfer function of the fast Fourier transforms for the sample and water reference data sets.Values for vL are determined from  ?= /&(  @ ⁄ ) − [(1 2 ⁄ )(Δ Δ ⁄ )]( where L is the material thickness and vW = 1,492 m/s the water sound speed.We determine vL = 1,480 m/s ± 10 m/s for the 4 mm-thick Uralite, vL = 1,498 m/s ± 5 m/s for the undoped gel sample (both the Uralite and Carbopol 2050 gel are closely impedance-matched to water), vL = 1,306 m/s ± 7 m/s (f < 660 kHz) and vL = 1,398 m/s ± 14 m/s (f > 660 kHz) for the sample with f = 0.21%, vL = 1,052 m/s ± 5 m/s (f < 702 kHz), vL = 670 m/s ± 7 m/s (702 kHz < f < 759 kHz), and vL = 444 m/s ± 9 m/s (f > 759 kHz) for the sample with f = 0.81%, vL = 874 m/s ± 4 m/s (f < 700 kHz) and vL = 398 m/s ± 3 m/s (f > 700 kHz) for the sample with f = 1.58%, vL = 742 m/s ± 4 m/s (f < 665 kHz) and vL = 307 m/s ± 6 m/s (f > 665 kHz) for the sample with f = 1.95%, and vL = 791 m/s ± 5 m/s (f < 554 kHz) and vL = 427 m/s ± 2 m/s (f > 554kHz) for the sample with f = 2.55%.These vL values are for individual speckles, and a measure of the spread in vL across speckles is shown in Fig. 4 of the main text.From measurements of vL we then compute l using the standard expression l = vL/f.Additionally, we find lS decreases with increasing f across the range of EMB resonance frequencies targeted by the experiments (specifically, 360-800 kHz) to values considerably less than the doped gel thickness LG = 10 mm.As an example, at the frequencies where lS/l = 1 for f = 0.21% and 0.81% (661 kHz and 577 kHz for f = 0.21%, and 0.81%, respectively) we measure lS = 1.57mm and lS = 1.47 mm for f = 0.21%, and 0.81%, respectively.Supplementary Note 4: Viscous penetration depth and the Biot theory for long-wavelength sound propagation For 360-800 kHz, the viscous penetration depth d is always less than the EMB equilibrium diameter DO where  = (2  ⁄  $ ) & " ⁄ and h and rl are the gel viscosity and density, respectively, and w = 2pf is the angular frequency.For a pH neutral Carbopol ETD 2050 gel (the suspending gel used in our experiments) the viscosity is expected to be h ~ 10,000 mPa*s based upon the material's Technical Data Sheet, and we adopt this h value in our determination of d.Supplementary Fig. 6 shows the frequency spectra for the EMB equilibrium diameter DO and the viscous penetration depth d.That d is always less than DO across the full frequency range of interest suggests d is not an order parameter for the transition into the quasi-gaseous phase as described within the main text.Because d is always less than DO we can rule out the Biot theory for long-wavelength sound propagation in a porous medium [S8, S9], which considers mode decoupling and sound propagation primarily through the inhomogeneous fluid when d becomes less than the pore size: the Biot theory cannot explain an abrupt decrease in vL at fC* because we do not observe d become less than a critical parameter at this frequency.

Fig. S2
Fig.S2Predicted EMB scattering cross section s(f) versus frequency f for an equilibrium diameter DO = 90 µm, which is normalized to the equilibrium cross section pRO 2 where RO is the EMB equilibrium radius.The prediction accounts for acoustic radiation damping.On resonance, the effective scattering cross section is several orders of magnitude larger than the EMB equilibrium geometrical cross section.Inset: damping coefficient b as a function of f for damping from acoustic radiation, EMB shell viscosity, and thermal effects; damping is primarily due to acoustic radiation within the frequency range of interest (400-800 kHz).

Fig. S3
Fig. S3 Density of states nw versus the square of the resonance angular frequency wO for five different EMB equilibrium volume fractions f for which data is presented within the main text.Note, the nw versus wO 2 distribution is non-Gaussian while the EMB size distribution shown in the main text's Fig. 1c is Gaussian.

Fig. S4
Fig. S4 Water sound speed versus temperature T measured for two separate runs (solid blue circles and open violet squares, respectively) using the experimental setup shown in the main text's Fig. 1f.The sound speed is determined through a time-of-flight measurement for a 1 microsecond-wide impulse, and confirmed with a wavepacket similar to the one shown in the main text's Fig. 2a.The dashed red line is the standard published temperature-dependent sound speed profile for distilled water (see Supplementary Reference S7).

Fig. S5
Fig. S5 Measured change in phase angle Dq versus frequency f for an EMB equilibrium volume fraction f = 1.58%.The solid black circle data set is the same data set shown in Fig. 3a of the main text.The solid red line data set corresponds to the same data, but with an additional 182 µs of time added onto the time-windowed data following the wavepacket arrival.The agreement between the two data sets indicates the results are not a result of insufficient data or signal processing.

Fig. S6
Fig. S6 Frequency spectra for the EMB equilibrium diameter DO (solid black circles, left vertical axis) and viscous penetration depth d (dashed green line, right vertical axis).

Fig. S7
Fig. S7 Density of states nw versus excitation frequency f for five different EMB equilibrium volume fractions f for which data is presented within the main text.For each data set, the vertical dashed line indicates the critical frequency fC* at which a transition is observed between fluid-like and gaseous-like behavior as described within the main text.

Fig. S8
Fig. S8 Frequency spectrum, determined from the transmitted coherent wave, for the ratio of the scattering mean free path lS to effective wavelength l for the sample with an EMB equilibrium volume fraction f = 0.21%.The horizontal axis is plotted over the EMB resonance frequency range targeted by the experiments.For clarity, error bars (determined from uncertainties in the measured sound level and doped gel thickness) are shown for every 5th data point.The dashed green line is a reference to lS/l = 1.The shaded gray region highlights the frequency range over which we measure lS/l £ 1.The dashed pink line is an independent scattering approximation (ISA) prediction based upon Eq.S20.

Fig. S9
Fig. S9 Predicted EMB on-resonance effective diameter Deff (open blue triangles) for a single, isolated EMB and the corresponding equilibrium diameter DO (solid black squares) versus frequency f.

Fig. S10
Fig. S10 Predicted EMB on-resonance effective diameter Deff (open black circles) versus frequency f for a single, isolated EMB along with a fit to a power model, which highlights the inverse relationship between Deff and f.

Fig. S11
Fig. S11 Predicted EMB effective volume fraction feff versus frequency f for an EMB equilibrium volume fraction f = 0.21% (solid black circles).The shaded pink region highlights the range of f over which the independent scattering approximation (ISA) is qualitatively valid, which is determined from the fit shown in Supplementary Fig. S8.The shaded gray region highlights the frequency range where we measure lS/l £ 1.

Fig. S12
Fig. S12 Reflected sound level (SL) versus frequency f for an undoped 4 mm-thick Uralite sample (open wine triangles) and for four samples with EMB equilibrium volume fractions f = 0.81% (solid green triangles), f = 1.58% (solid black circles), f = 1.95% (open blue squares), and f = 2.55% (solid orange triangles).Here, the Uralite polymer thickness is the same thickness as the undoped Uralite pocket into which the doped gel is poured for the in-water measurements.

Fig. S14
Fig. S14 Incoherent wave analysis across the two asymptotic regimes: fluid-like (including diffusive states) and quasi-gaseous (f > fC*; in (c)-(e) the 750-800 kHz frequency range corresponds to the quasi-gaseous phase).The normalized transmitted intensity peak envelope I/IO (on a semi-logarithmic plot) is plotted versus time t for the incoherent energy and for the three doped samples for which data is presented in Fig. 2 of the main text: f = 1.58% in (a)-(c), f = 1.95% in (d), and f = 2.55% in (e).Incoherent wave data is digitally filtered to target those frequency ranges shown in each figure part based upon the lS/l data shown in Fig. 2(d)-(f) of the main text.I/IO is found from averaging over 11 different speckle measurements.Normalization is done so the input pulse peak is unity, and then so that in each figure part the maximum occurs at I/IO = 1.The time ranges are shifted from the experiment time so the maximum in I/IO occurs shortly after t = 0 s.The solid red line in (a) is a linear fit (diffusion) with the characteristic diffusion time tD serving as a free fitting parameter.Dashed magenta lines in (b), (d), and (e) are fits to the localization self-consistent theory (SCT) with DB, x, and ta (bare diffusion coefficient, localization length, and characteristic absorption time, respectively) serving as free fitting parameters.

Fig. S15
Fig. S15 Wave localization effects (and SCT analysis) in EMB-doped gel for when lS/l < 1. Normalized transmitted intensity peak envelope I/IO (on a semi-logarithmic plot) plotted versus time t for the incoherent energy and for the two doped samples that represent the full range of f over which the behavior of fC* versus f is studied within the main text (Fig. 5).Incoherent wave data is digitally filtered to target those frequency ranges shown in each figure part where lS/l < 1. I/IO is found from averaging over 11 different speckle measurements.Normalization is done so the input pulse peak is unity.Data shown in (b) is the same data set shown in Supplementary Fig. 14(e).The time ranges are shifted from the experiment time so the maximum in I/IO occurs shortly after t = 0 s.Solid red lines are fits to the self-consistent theory (SCT) of localization with DB, x, and ta (bare diffusion coefficient, localization length, and characteristic absorption time, respectively) serving as free fitting parameters, and the resultant values from the SCT fits are shown in red.Dashed purple lines correspond to setting ta = 10 µs in the SCT fit while keeping all other parameters fixed at the values specified.